Robust Superradiance and Spontaneous Spin Ordering in Disordered Waveguide QED

TL;DR

This work addresses whether Dicke superradiance survives strong disorder in a disordered 1D waveguide QED array. It combines scalable semiclassical methods (DTWA and QSDMF) with analytic product-state bounds to show that the characteristic $N^2$ scaling of the peak emission rate and the $t_{\star}\sim\log(N)/N$ burst time persist in the presence of strong spatial and spectral disorder, albeit with disorder-dependent finite-size corrections. A central finding is disorder-induced spontaneous spin ordering, where dipoles align along locally defined directions determined by their positions, enabling constructive interference and mirror-asymmetric photon correlations. The results hold under moderate non-Markovianity and inhomogeneous broadening, offering a robust framework for understanding collective emission in realistic disordered quantum optical systems and informing the design of robust superradiant devices.

Abstract

We study the collective emission of a disordered array of $N$ excited two-level atoms into a one-dimensional photonic waveguide. In the perfectly ordered case, where atoms are spaced by exact integer multiples of the wavelength, the system exhibits the characteristic superradiant burst with a peak emission rate scaling as $N^2$. Using large-scale semiclassical simulations, we find that this key signature of superradiance remains asymptotically robust under strong spatial and spectral disorder, but also exhibits subtle finite-size scaling toward this limit. To explain our observations, we provide an analytical variational estimate for the maximal decay rate, which tightly bounds the numerical results and reveals how disorder shapes the collective decay. Specifically, we find that even in the presence of strong disorder, the spins tend to self-organize spontaneously according to their locations, which overall optimizes constructive interference effects and explains the emergence of mirror-asymmetric correlations in superradiant decay. These findings resolve important open questions regarding the existence and nature of superradiance in strongly disordered arrays and offer valuable insights for understanding collective quantum optical phenomena in realistic systems.

Figures (20)

• Figure 1: (a) Sketch of a disordered waveguide QED system, where $N$ excited two-level atoms are located at random positions $z_j=j\times \lambda_0 +\delta_j$ along a 1D photonic channel and undergo collective spontaneous decay. As shown in the inset, the displacements $\delta_j$ from exact multiples of the resonant wavelength $\lambda_0$ are chosen randomly from the interval $[-\Theta/2,\Theta/2)$ to interpolate between a regular array ($\Theta=0)$ and a uniformly distributed gas of atoms ($\Theta\gg1$). (b) Illustration of the two ordering patterns (red and blue), which are generated spontaneously between the atomic dipoles during the superradiant decay. The dipoles self-align according to their distances, with $\phi_{j} \approx \phi_{1} + k_{0} (z_{j} - z_{1})$ (blue arrows) or $\phi_{j} \approx \phi_{1} - k_{0} (z_{j} - z_{1})$ (red arrows), which even for strong disorder yields constructive interference and enhanced collective emission either to the left or to the right.
• Figure 2: Numerical methods. (a) Sketch of the extended system described by Eq. \ref{['extendedME']}, where the collective decay into left- and right-going waveguide modes is realized effectively via the coupling of the atoms to two lossy cavity modes with decay rate $\kappa$. This extended model underlies the DTWA method discussed in Sec. () and is used for the study of non-Markovian effects in Sec. \ref{['secNonMarkov']}. In the Markovian limit $\kappa/(\gamma N) \to \infty$, the model recovers the original Lindblad master equation in Eq. \ref{['extendedME']}. (b) Sketch of a homodyne measurement setup that corresponds to the QSD unraveling of the master equation, as discussed in Sec. \ref{['secQSD']}. Trajectories obtained in our QSDMF simulations represent (factorized) states of the atoms that are conditioned on a given history of the measured homodyne currents $I_{\rm R}(t)$ and $I_{\rm L}(t)$.
• Figure 3: Benchmarking of DTWA and QSDMF methods. (a) and (c) Plot of the collective decay rate $\mathcal{R}(t)$ for $N = 5, 10, 15$ (bottom to top) and for a disorder strength of $\Theta = \pi$. The results obtained from exact quantum jump (QJ) simulations (solid lines) are compared with those from DTWA and QSDMF simulations (dashed lines) with $\mathcal{N}=10^{3}$ trajectories. Using the same methods, (b) and (d) show the corresponding error for the normalized decay rate $\mathcal{R}(t)/N^2$ when compared to QJ results.
• Figure 4: Superradiant dynamics for spatial disorder strength $\Theta = \pi$. (a) Decay of the average excited-state population $\mathcal{P}_{\rm e}$, showing an increasingly rapid decay as $N$ increases. (b) Plot of the corresponding decay rate per atom, $\mathcal{R}(t)/N$, which exhibits a sharp peak at the burst time $t_{\star} \sim \log(N)/N$. The inset shows the same date with a rescaled time. Both plots have been obtained using the DTWA method with averages taken over $\mathcal{N}=10^3$ trajectories.
• Figure 5: Superradiant scaling in the presence of spatial disorder. (a) Plot of the scaled superradiant rate, $\mathcal{R}_{\star}/(\gamma N^2)$, as a function of the atom number $N$ and for a varying disorder strength $\Theta$. The plot demonstrates the asymptotic recovery of the Dicke superradiant scaling with finite-size corrections that are fitted to the analytic prediction in Eq. \ref{['FiniteSizeFit']} (dashed lines). Similarly, the numerical data presented in (b) shows the corresponding scaled burst time, $t_{\star} \gamma N/\log(N)$, again converging to the asymptotic Dicke scaling for large $N$. In both plots, the results have been obtained using the DTWA method, averaged over $\mathcal{N}=10^3$ trajectories.